The ratio of maximum particle velocity to wave velocity is [ where symbols have their usual meanings ]

Options

A. kA
B. `A omega `
C. `k omega `
D. `(omega ) /k`

Answer

To solve the problem of finding the ratio of maximum particle velocity to wave velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The **maximum particle velocity (Pmax)** in a wave is given by the formula:
\[ P_{\text{max}} = A \cdot \omega \]
where \( A \) is the amplitude of the wave and \( \omega \) is the angular frequency.
- The **wave velocity (V)** is given by:
\[ V = \frac{\omega}{k} \]
where \( k \) is the angular wave number.

2. **Set Up the Ratio**:
- We need to find the ratio of maximum particle velocity to wave velocity:
\[ \text{Ratio} = \frac{P_{\text{max}}}{V} \]

3. **Substitute the Formulas**:
- Substitute the expressions for \( P_{\text{max}} \) and \( V \):
\[ \text{Ratio} = \frac{A \cdot \omega}{\frac{\omega}{k}} \]

4. **Simplify the Expression**:
- When we simplify the fraction, we can multiply by the reciprocal of the denominator:
\[ \text{Ratio} = A \cdot \omega \cdot \frac{k}{\omega} \]
- The \( \omega \) terms cancel out:
\[ \text{Ratio} = A \cdot k \]

5. **Final Result**:
- Thus, the ratio of maximum particle velocity to wave velocity is:
\[ \text{Ratio} = A \cdot k \]

### Conclusion:
The ratio of maximum particle velocity to wave velocity is \( A \cdot k \), where \( A \) is the amplitude and \( k \) is the angular wave number.